Physical Chemistry Third Edition

1068 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

The formula given in Eq. (25.4-11) applies to heteronuclear diatomic molecules, but

must be modified for homonuclear diatomic molecules, for which either odd values

ofJor even values ofJdo not occur, as discussed in Chapter 22. This restriction

corresponds to omitting half of the rectangles in Figure 25.4, approximately cutting the

value of the sum in half. For diatomic molecules thesymmetry numberσis defined in

Section 22.5.

σ

{

2 homonuclear diatomic molecules

1 heteronuclear diatomic molecules (25.4-12)

We can write for all diatomic molecules

zrot

1

σ

8 π^2 IekBT

h^2



1

σ

kBT

hBe



1

σ

kBT

hcB ̃e

(

diatomic

substances

)

(25.4-13)

In the translational partition function, the replacement of a sum by an integral gave an

excellent approximation. In the rotational partition function it is a very good approxima-

tion at ordinary temperatures for molecules containing heavy atoms, a good approxima-

tion for molecules containing atoms of moderate mass, and a fairly poor approximation

for molecules containing light atoms such as hydrogen. The approximation is better

for larger values of the rotational partition function, so it is a better approximation

at higher temperatures. If the formula in Eq. (25.4-13) gives a value smaller than 10,

this approximation might not be adequate. Partial sums of the series representing the

rotational partition function can be summed explicitly, and formulas with corrections

can also be used (see Problem 25.31).^5

EXAMPLE25.7

Calculate the rotational partition function for^35 Cl 2 at 298.15 K.

Solution

μ

0 .03497 kg mol−^1

2(6. 0221 × 1023 mol−^1 )

 2. 903 × 10 −^26 kg

From Table A.22 in Appendix A,re 1. 988 × 10 −^10 m.

Ie(2. 903 × 10 −^26 kg)(1. 988 × 10 −^10 m)^2  1. 147 × 10 −^45 kg m^2

Zrot

1

2

8 π^2 (1. 147 × 10 −^45 kg m^2 )(1. 3807 × 10 −^23 JK−^1 )(298.15 K)

(6. 6261 × 10 −^34 Js)^2

 424. 7

Exercise 25.18

Repeat the calculation of the rotational partition function of^35 Cl 2 at 298.15 K using the value

ofB ̃efrom Table A.22 in Appendix A.

(^5) N. Davidson,op. cit., p. 111ff (note 2); R. S. McDowell,J. Chem. Phys., 88 , 356 (1988).